Indices
Simplify the following:
Example 1
$$ x^4 × x^5 $$
Solution
$$ \because a^m × a^n = a^{m+n}$$
$$ \therefore x^4 × x^5 = x^{4+5} = x^9$$
Example 2
$$ y^3 × y^2 $$
Solution
$$ \because a^m × a^n = a^{m+n}$$
$$ \therefore y^3 × y^2 = y^{3+2} = y^5$$
Example 3
$$ c^6 × c$$
Solution
$$ \because a^m × a^n = a^{m+n}$$
$$ \therefore c^6 × c = c^6 × c^1 = c^{6+1} = c^7$$
Example 4
$$ x^9 ÷ x^7 $$
Solution
$$ \because a^m ÷ a^n = a^{m-n}$$
$$ \therefore x^9 ÷ x^7 = x^{9-7} = x^2$$
Example 5
$$ d^5 ÷ d^2 $$
Solution
$$ \because a^m ÷ a^n = a^{m-n}$$
$$ \therefore d^5 ÷ d^2 = d^{5-2} = d^3$$
Example 6
$$ 4a^2 × 2a^3 $$
Solution
$$ \because a^m × a^n = a^{m+n}$$
$$ \therefore 4a^2 × 2a^3 = 8a^{2+3} = 8a^5$$
Example 7
$$ 7x^4 × x^8 $$
Solution
$$ \because a^m × a^n = a^{m+n}$$
$$ \therefore 7x^4 × x^8 = 7x^{4+8} = 7x^{12}$$
Example 8
$$ 16a^6 ÷ 8a^5 $$
Solution
$$ \because a^m ÷ a^n = a^{m-n}$$
$$ \therefore 16a^6 ÷ 8a^5 = 2a^{6-5} = 2a^1 =2a$$
Example 9
$$ 49b^7 ÷ 7b^3 $$
Solution
$$ \because a^m ÷ a^n = a^{m-n}$$
$$ \therefore 49b^7 ÷ 7b^3 = 7b^{7-3} = 7b^4$$
Example 10
$$ (2c^4)^3 $$
Solution
$$ \because (ab)^n = a^n b^n \; \textup{and} \; (a^m)^n = a^{m×n}$$
$$ \therefore (2c^4)^3=(2)^3(c^4)^3=8c^{4×3}=8c^{12}$$
Example 11
$$ (3d^7)^2 $$
Solution
$$ \because (ab)^n = a^n b^n \; \textup{and} \; (a^m)^n = a^{mn}$$
$$ \therefore (3d^7)^2= (3)^2(d^7)^2=9d^{7×2}=9d^{14}$$
Example 12
$$ (5x^3 × 2x^4)^3$$
Solution
$$ \because a^m × a^n = a^{m+n} ,\; (ab)^n = a^n b^n $$
$$\textup{and} \; (a^m)^n = a^{m×n}$$
$$ \therefore (5x^3 × 2x^4)^3 $$
$$ = (10x^{3+4})^3 $$
$$ = (10x^7)^3 $$
$$ = (10)^3(x^7)^3 $$
$$ = 1000x^{7×3} $$
$$ = 1000x^{21} $$
Example 13
$$ (6a^2 × 2a^6)^2$$
Solution
$$ \because a^m × a^n = a^{m+n}, \; (ab)^n = a^n b^n$$
$$\textup{and} \; (a^m)^n = a^{m×n}$$
$$ \therefore (6a^2 × 2a^6)^2 $$
$$ = (12a^{2+6})^2 $$
$$ = (12a^8)^2 $$
$$ = (12)^2(a^8)^2 $$
$$ = 144a^{8×2} $$
$$ = 144a^{16} $$
Example 14
$$ (8y^9 ÷ 2y^6)^4$$
Solution
$$ \because a^m ÷ a^n = a^{m-n}, \; (ab)^n = a^n b^n$$
$$\textup{and} \; (a^m)^n = a^{m×n}$$
$$ \therefore (8y^9 ÷ 2y^6)^4 $$
$$ = (4y^{9-6})^4 $$
$$ = (4y^3)^4 $$
$$ = (4)^4(y^3)^4 $$
$$ = 256a^{3×4} $$
$$ = 256a^{12} $$
Example 15
$$ (6z^8 ÷ 3z^4)^3$$
Solution
$$ \because a^m ÷ a^n = a^{m-n}, \; (ab)^n = a^n b^n$$
$$ \textup{and} \; (a^m)^n = a^{m×n}$$
$$ \therefore (6z^8 ÷ 3z^4)^3 $$
$$ = (2z^{8-4})^3 $$
$$ = (2z^4)^3 $$
$$ = (2)^3(z^4)^6 $$
$$ = 8z^{4×6} $$
$$ = 8z^{24} $$
Example 16
$$ 2x^2y^3 × 3x^4y^2$$
Solution
$$ \because a^m × a^n = a^{m+n} $$
$$ \therefore 2x^2y^3 × 3x^4y^2 $$
$$ = 6x^{2+4}y^{3+2}$$
$$ = 6x^6y^5$$
Example 17
$$ 4c^4d^5 × 6c^2d^3$$
Solution
$$ \because a^m × a^n = a^{m+n} $$
$$ \therefore 4c^4d^5 × 6c^2d^3 $$
$$ = 24c^{4+2}d^{5+3}$$
$$ = 24c^6d^8$$
Example 18
$$ 25a^5b^6 ÷ 5a^2b^4$$
Solution
$$ \because a^m ÷ a^n = a^{m-n} $$
$$ \therefore 25a^5b^6 ÷ 5a^2b^4 $$
$$ = 5a^{5-2}b^{6-4}$$
$$ = 5a^3b^2$$
Example 19
$$ 48e^7f^8 ÷ 8e^3f^2$$
Solution
$$ \because a^m ÷ a^n = a^{m-n} $$
$$ \therefore 48e^7f^8 ÷ 8e^3f^2 $$
$$ = 6e^{7-3}f^{8-2}$$
$$ = 6e^4f^6$$
Example 20
$$ (3x^3y^4)^2 × 2x^8y^4$$
Solution
$$ \because (ab)^n = a^n b^n, \; (a^m)^n = a^{m×n}$$
$$ \textup{and} \; a^m × a^n = a^{m+n} $$
$$ \therefore (3x^3y^4)^2 × 2x^8y^4 $$
$$ = (3)^2(x^3)^2(y^4)^2 × 2x^8y^4 $$
$$ = 9x^{3×2}y^{4×2} × 2x^8y^4 $$
$$ = 9x^6y^8 × 2x^8y^4 $$
$$ = 18x^{6+8}y^{8+4}$$
$$ = 18x^{14}y^{12}$$